Question

Answer 1

\(A=\dfrac{3yz}{x}+\dfrac{4xz}{y}+\dfrac{5xy}{z}=\left(\dfrac{yz}{x}+\dfrac{xz}{y}\right)+2\left(\dfrac{yz}{x}+\dfrac{xy}{z}\right)+3\left(\dfrac{xz}{y}+\dfrac{xy}{z}\right)\\ \Leftrightarrow A\ge2\sqrt{\dfrac{xyz^2}{xy}}+2\cdot2\sqrt{\dfrac{xy^2z}{xz}}+3\cdot2\sqrt{\dfrac{x^2yz}{yx}}\\ \Leftrightarrow A\ge2z+4x+6y\\ \Leftrightarrow A\ge4\left(x+y\right)+2\left(y+z\right)\ge4\cdot2\sqrt{xy}+2\cdot2\sqrt{yz}\\ \Leftrightarrow A\ge4\left(2\sqrt{xy}+\sqrt{yz}\right)=4\cdot1=4\)

Dấu \("="\Leftrightarrow x=y=z=\dfrac{1}{3}\)